What is the distribution of the sum of squared chi-square random variables?

Question

What would be the distribution of the following equation:

$$y = a^2 + 2ad + d^2$$

where $a$ and $d$ are independent non-central chi-square random variables with $2 \textbf{M}$ degrees of freedom.

OBS.: The r.v.'s generating both $a$ and $d$ have $\mu = 0$ and $\sigma^2 \neq 1$, let's say $\sigma^2 = c$.

I've just added some more information to the question. They are non-central chi-square r.v.'s as they were generated by non-standard circular symmetric complex Gaussian random variables. — Felipe Augusto de Figueiredo, Dec 27 '16 at 19:45
@Alecos Papadopoulos, yes, each one of them has $2M$ degrees of freedom. — Felipe Augusto de Figueiredo, Dec 27 '16 at 22:02
Your last line beginning "OBS" appears to contradict everything else you have written. Could you clarify what you're trying to say? Specifically, exactly how are "the r.v.'s generating both $a$ and $d$" supposed to "generate" them? — whuber, Dec 27 '16 at 22:07
@whuber, both $a$and $d$ are generated by the sum $\textbf{M}$ of non-standard circular symmetric complex Gaussian random variables. My Observation (OBS) aimed to highlight that fact, that those r.v.'s which are summed to generate $a$ and $d$ don't have $\mu = 0$ and $\isigma^2 = 1$. I hope this answers your question. — Felipe Augusto de Figueiredo, Dec 27 '16 at 22:24
Felipe, in your question you state $a$ and $d$ do "have $\mu=0$" but now in your latest comment you state they don't have this property. Which is it?? — whuber, Dec 27 '16 at 23:23
@whuber, sorry but you did not understand. $a$ and $d$ are non-central chi-square r.v.'s which are generated by the sum of $\textbf{M}$ non-standard circular symmetric complex Gaussian random variables with $\mu = 0$ and $\sigma = c$. I haven't stated $a$ and $d$ have $\mu = 0$. Please, read again. Thanks! — Felipe Augusto de Figueiredo, Dec 28 '16 at 12:06
Thank you for trying to explain, but I still cannot make sense of it. Where you write "$a$ and $d$ are independent non-central chi-square random variables" it sounds like you are summing squares of Normal random variables that have nonzero means, because that's how non-central Chi-squared variables usually arise. But later your write "The r.v.'s generating both $a$ and $d$ have $\mu=0$", which suggests you are working with central Chi-squared variables. I suspect these are the inconsistencies that prompted the initial comment by @Glen_b. Could you show explicitly what $a$ and $d$ are? — whuber, Dec 28 '16 at 14:55

score 7 · Accepted Answer · edited Apr 13 '17 at 12:44

7

If $a, d\sim\chi^2_{2M}$ are independent, then $X=a+d$ will have $\chi^2_{4M}$ distribution. Since $X$ is non-negative, CDF of $Y=a^2+2ad+d^2=(a+d)^2=X^2$ can be found by noting $$F_Y(y)=P(Y\leq y)=P(X^2\leq y)=P(X\leq \sqrt{y})=F_X(\sqrt{y}).$$ Therefore, $$f_Y(y)=\frac{1}{2\sqrt{y}}f_X(\sqrt{y})=\frac{1}{2^{2M+1}\Gamma(2M)}y^{M-1}e^{-\sqrt{y}/2}.$$

If $a$ and $d$ are correlated then things are much more intricate. See for example N. H. Gordon & P. F. Ramig's Cumulative distribution function of the sum of correlated chi-squared random variables (1983) for a definition of multivariate chi-squared and distribution of its sum.

If $\mu\neq 2M$ then you are dealing with non-central chi-squared so the above will no longer be valid. This post may provide some insight.

EDIT: Based on the new information it seems $a$ and $d$ are formed by summing up normal r.v. with non-unit variance. Recall if $Z\sim N(0, 1)$ then $\sqrt{c}Z\sim N(0, c)$. Since now $$a=c\sum_{i=1}^{2M}Z_i^2=d,$$ both $a,d$ will have chi-squared distribution scaled by $c$, i.e. $\Gamma(M, 2c)$ distribution. In this case $X=a+d$ will be $\Gamma(2M, 2c)$ distributed. As a result, for $Y=X^2$ we have $$f_Y(y)=\frac{1}{2(2c)^{2M}\Gamma(2M)}y^{M-1}e^{-\sqrt{y}/2c}.$$

edited Apr 13 '17 at 12:44

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answered Dec 26 '16 at 23:50

Francis

3,150

How does mu enter? Is it suppose to be the mean of one of the chi-square variables? I suspect it has nothing to do with the problem. – Michael R. Chernick Dec 27 '16 at 00:01
@MichaelChernick: probably means $a, d$ can be non-central chi-squared? – Francis Dec 27 '16 at 00:06
I suppose you can make that assumption but the OP does not make any connection. I think you took the right approach, the non-central could not enter into this problem. X is the square of a chi-square here. In the case of independence that you used here what is the this distribution called? – Michael R. Chernick Dec 27 '16 at 00:19
@MichaelChernick I am not sure if there is a special name associated with the distribution. "chi-tesseracted" maybe? – Francis Dec 27 '16 at 00:36
$a$ and $d$ are non-central chi-squared. – Felipe Augusto de Figueiredo Dec 27 '16 at 19:46
@Francis, when do you mean non-central chi-squared, are you considering standard circular symmetric complex Gaussian random variables? I mean, with $\mu = 0$ and $\sigma = I$? – Felipe Augusto de Figueiredo Dec 27 '16 at 19:59
@Francis, what happens if the r.v.'s forming the chi-square r.v., $a$ and $d$, have $\mu = 0$ and $\sigma = cM$? – Felipe Augusto de Figueiredo Dec 27 '16 at 21:04
@FelipeAugustodeFigueiredo: non-central chi-squared are formed by summing squared normal rv with unit variance and mean other than 0. Summing up non-unit variance normal rv will give you a scaled chi-squared rv on the other hand. – Francis Dec 28 '16 at 04:42
@Francis, thank you very much for helping me with that! Could you, please, have a look at the following question and see you you can help me? http://stats.stackexchange.com/questions/253607/what-is-the-distribution-of-the-ratio-of-independent-chi-square-random-variables – Felipe Augusto de Figueiredo Dec 28 '16 at 21:59

score 4 · Answer 2 · edited Apr 13 '17 at 12:44

4

Since a non-central chi-square is a sum of independent rv's, then the sum of two independent non-central chi-squares $X = a+b$ is also a non-central chi-square with parameters the sum of the corresponding parameters of the two components, $k_x = k_a+k_b$ (degrees of freedom), $\lambda_x = \lambda_a+\lambda_b$ (non-centrality parameter).

To obtain the distribution function of its square $Y =X^2$ , one can apply the "CDF method" (as in @francis answer),

$$F_Y(y)=P(Y\leq y)=P(X^2\leq y)=P(X\leq \sqrt{y})=F_X(\sqrt{y})$$

and where

$$F_X(x)=1 - Q_{k_x/2} \left( \sqrt{\lambda_x}, \sqrt{x} \right)$$

so

$$F_Y(y)=1 - Q_{k_x/2} \left( \sqrt{\lambda_x}, y^{1/4} \right)$$

where $Q$ here is Marcum's Q-function.

The above apply to non-central chi-squares formed as sums of independent squared normals each with unitary variance but different mean.

ADDENDUM RESPONDING TO QUESTION'S EDIT

If the base rv's are $N(0,c)$, then the square of each is a $Gamma (1/2,2c)$ see https://stats.stackexchange.com/a/122864/28746 .

So the rv $a \sim Gamma (M, 2c)$ and $b \sim Gamma (M, 2c)$ so also $X = a+b \sim Gamma(2M, 2c)$ (shape-scale parametrization, and see the wikipedia article for the additive properties for Gamma).

Then one can apply again the CDF method to find the CDF of the square $Y = X^2$

edited Apr 13 '17 at 12:44

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answered Dec 27 '16 at 20:14

Alecos Papadopoulos

58,953

@FelipeAugustodeFigueiredo Sorry, I am not familiar with complex rv's. My answer took as given the fact that we start from non-central chi-squares. – Alecos Papadopoulos Dec 27 '16 at 20:34
What if the r.v.'s are circular symmetric complex Gaussian random variables with $\mu = 0$ and $\sigma = cI$? – Felipe Augusto de Figueiredo Dec 27 '16 at 20:35
let's forget about complex rv's. What if the r.v.'s generating $a$ and $d$ are Gaussian r.v.'s with $\mu = 0$ and $\sigma \neq 1$? All Gaussian r.v.'s have the same variance, let's call it $c$. – Felipe Augusto de Figueiredo Dec 27 '16 at 21:28
could you please help me with the following question: http://stats.stackexchange.com/questions/253764/expectation-of-the-ratio-between-beta-and-gamma-random-variables. Any hint would be very appreciated. Thanks! – Felipe Augusto de Figueiredo Dec 30 '16 at 12:41
@FelipeAugustodeFigueiredo I am afraid I do not have something to offer for that question. – Alecos Papadopoulos Dec 30 '16 at 18:58

What is the distribution of the sum of squared chi-square random variables?

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